Computación y Sistemas Vol. 9 Núm. 4, pp. 355-369
 2006, CIC-IPN, ISSN 1405-5546, Impreso en México
Two Robust Techniques for Segmentation of Biomedical Images
Dos Técnicas Robustas para la Segmentación de Imágenes Biomédicas
Roberto Rodríguez, Patricio J. Castillo, Valia Guerra1, Ana G. Suárez and Ebroul Izquierdo2
Digital Signal Processing Group
1Numerical Methods Group
Institute of Cybernetics, Mathematics & Physics (ICIMAF)
2Department of Electronic Engineering, Queen Mary, London University
rrm@icmf.inf.cu, vguerra@icmf.inf.cu, anagloria@icmf.inf.cu
Article received on October 15, 2004; accepted on August 18, 2006
Abstract
Image segmentation plays an important role in many systems of computer vision. According to criterions of many
authors the segmentation finishes when it satisfies the goals of the observer. For that reason, an only method there is
not able of solving all the problems that exists in the present time. In this work, we carry out a comparison between
two segmentation techniques; namely, through the mean shift, where we give a new algorithm, and by using spectral
methods. In the paper we discuss, through examples with biomedical real images, the advantages and disadvantages
of them.
Keywords: Image segmentation, mean shift, spectral methods.
Resumen
La segmentación de imagen juega un importante rol en muchos sistemas de visión por computadora. Acorde al
criterio de muchos autores la segmentación finaliza cuando son satisfechos los objetivos del observador. Por esa
razón, no hay un único método capaz de resolver todos los problemas que existen en la época actual. En este trabajo,
nosotros llevamos a cabo una comparación entre dos técnicas de segmentación; a saber, a través de la media
desplazada, donde nosotros ofrecemos un nuevo algortimo, y usando los métodos espectrales. En el artículo nosotros
discutimos, a través de ejemplos con imágenes biomédicas reales, las ventajas y desventajas de ellos.
Palabras claves: Segmentación de imagen, media desplazada, métodos espectrales
1 Introduction
Segmentation and contour extraction are important steps in many systems of high level. Segmented images are now used
routinely in a multitude of different applications, such as, diagnosis, treatment planning, localization of pathology, study
of anatomical structure, computer-integrated surgery, among others. However, image segmentation remains a difficult
task due to both the variability of object shapes and the variation in image quality. Particularly, medical images are often
corrupted by noise and sampling artifacts, which can cause considerable difficulties when applying rigid methods.
Nevertheless, in spite of the most complex algorithms developed until present, segmentation continues being very
dependent on the application and an only method there is not able of solving all the problems arising in the universe.
The pathological anatomy is a speciality, where the use of different techniques of digital image processing (DIP)
allows improving the accuracy of the diagnosis of many diseases. One of the most important diseases in order to study
today is the cancer, which constitutes one of the first causes of death in all those countries where the epidemics do not have
an important weight.
Many segmentation methods have been proposed for medical-image data [1-6]. Unfortunately, segmentation using
traditional low-level image processing techniques, such as thresholding, histogram, region growing and other classical
operations requires a considerable amount of interactive guidance in order to attain satisfactory results. Automating
these model-free approaches is difficult because of complexity, shadows, and variability within and across individual
355
Roberto Rodríguez, et al.
objects. Furthermore, noise and other image artifacts can cause incorrect regions or boundary discontinuities in objects
recovered from these methods.
A robust technique used in the present work was the mean shift, which is a nonparemetric procedure and it is an
extremely versatile tool for feature analysis and can provide reliable solutions for many vision tasks [7, 8]. The mean
shift was proposed in 1975 by Fukunaga and Hostetler [9] and largely forgotten until Cheng´s paper [10] rekindled
interest in it. In spite of its excellent qualities, the mean shift procedure does not seem to be known in statistical
literature.
The other technique used to carry out the comparison was the spectral methods, which is based on the calculation of
the eigenvectors and eigenvalues of a matrix obtained from the affinities of the pixels in the image. In particular, it was
used a spectral method called of Normalized Cut [11], which measures both total dissimilarity between the different
groups as well as the total similarity within groups. The main idea is solving the eigenvalue problem for a small random
subset of pixels (sample) and then extrapolating this solution to the full set of pixels. The theoretical aspects of the used
methods are exposed in the developing of the work.
The paper is structured of the following way: in section 2 are given the details of the most significant theoretical
aspects of each of the used methods. In section 3 the characteristics of the study images are described, and in section 4
the steps in the experimentation are explained. Here, we will carry out an analysis and discussion of both methods.
Finally, in section 5 the most important conclusions are given.
2 Theoretical Aspects
2.1 The mean shift
Kernel density estimation (know as the Parzen window technique in pattern recognition literature [8]) is the most
popular density estimation method. Given n data points xi i=1, …., n in the d-dimensional space Rd , the multivariate
kernel density estimator with kernel K (x) and a symmetric positive definite dxd bandwidth matrix H, computed in the
point x is given by
fˆ x  
1
n
 K x  x 
H
n i 1
i
(1)
where,
K H x   H
1
2
 1 
K  H 2 x 


(2)
Some considerations on matrix H and the mathematical background and other definitions can be found in [8].
Employing only one bandwidth parameter, the kernel density estimator in the expression (1) becomes the well-known
expression
1
fˆ ( x)  d
nh
n
 x  xi 

h 
 K 
i 1
(3)
The quality of a kernel density estimator is measured by the mean of the square error between the density and its
estimate, integrated over the domain of definition. In practice, however, only an asymptotic approximation on this
measure can be computed [8].
A digital image can be represented as a two-dimensional array of p-dimensional vectors (pixels), where p =1 in the gray
level case, three for color images, and p > 3 in the multispectral case [8]. The space of the pixels is known as the spatial
domain, while the gray level, color, or spectral information is represented in the range domain. For both domains,
Euclidean metric is assumed. In this edition the case of gray level images will be only analyzed.
356
Two Robust Techniques for Segmentation of Biomedical Images
As was pointed in [8] when the location and range vectors are concatenated in the joint spatial-range domain of
dimension d = p+2, their different nature has to be compensated by proper normalization with the hs and hr parameters.
Thus, the multi-variable kernel is defined as the product of two radially symmetric kernels and the Euclidean metric
allows a single bandwidth for each domain, that is,
2
2
s
r
 x

K h , h (x)  2 p k 
s r
h s h r  hs
  x
k 
  hr
 
C




were xs is the spatial part, xr is the range part of a feature vector, k (x) the common profile used in both domains, hs and
hr the employed kernel bandwidths, and C the corresponding normalization constant.
The novelty lies in applying the mean shift procedure for the data points in the joint spatial-range domain. Each data
point becomes associated to a point of convergence which represents the local mode of the density in the d-dimensional
space. The process, having the parameters hs and hr, take into account simultaneously both the spatial and range
information. For the segmentation task, the convergence points sufficiently close in the joint domain are fused to obtain
the homogenous regions in the image.
2.1.1 Mean shift as normalized density gradient estimate
The iterative procedure of the mean shift is introduced as normalized density gradient estimate. By employing a
differentiable kernel, an estimate of the density gradient can be defined as the gradient of the kernel density estimate;
that is,
ˆ f(x)  fˆ(x) 

n
x x
 K( h i )
nh d i 1
1
(4)
Conditions on the kernel K(x) and the window radio h to guarantee asymptotic unbiasedness, mean-square
consistency, and uniform consistency of the estimate in the expression (4) are derived in [9].

1/2 c d 1 (d  2 ) (1 - x 2 ) if x  1
For example, for Epanechnikov kernel ( K E (x)  

otherwise
0
)
where cd is the volume of the d-dimensional sphere [8].
Then, for the Epanechnikov kernel the density gradient estimate in the expression (4) becomes,
ˆ f ( x) 

E
where the region
nx
1
d 2
d  2  1

(
x

x
)


 i
n(h d cd ) h 2 xi Sh ( x )
n(h d cd ) h 2  nx


(
x

x
)

i

xi S h ( x )

(5)
Sh (x) is a hypersphere of radius h having the volume h d cd centered on x, and containing n x data
points, that is, the uniform kernel [7, 8]. The last term in expression (5) is called the sample mean shift,
M h,U (x) 
1
nx
1
(xi  x) 
xi  x


n
x S (x)
x S (x)
i
x
h
357
i
h
(6)
Roberto Rodríguez, et al.
As it will be shown in section 2.1.1.1, using a kernel different form the Epanechnikov kernel for the derivation of the
density gradient estimate results in a weighted mean computation in the expression (6).
The quantity
nx
is the kernel density estimate fˆU ( x) computed with the hypersphere S h (x) , and thus we
n ( h d cd )
can write the expression (5) as,
d 2
ˆ f E ( x )  fˆU ( x )  2 M h, U ( x )
h
(7)
ˆ f ( x)
h2 
E
M h , U ( x) 
d  2 fˆU ( x)
(8)
which yields,
The expression (5) shows that an estimate of the normalized gradient can be obtained by computing the sample mean
shift in a uniform kernel centered on x. In addition, when this estimate is obtained with the Epanechnikov kernel, the
mean shift vector has the direction of the gradient of the density estimate at x. As the mean shift vector always points
towards the direction of the maximum increase in the density, it can define a path leading to a local density maximum;
that is, to a mode of the density (see Fig. 1).
f (x)
x2 x3
x1
x4
x5
x6
x7
Fig. 1. Gradient mode clustering.
From expression (8) one can deduce that the normalized gradient introduces a desirable adaptive behavior, since the
mean shift step is large for low density regions corresponding to valleys, and decreases as x approaches a mode.
Mathematically, this is justified since
ˆ f (x)

E
ˆ f (x) . Thus the corresponding step size for the same gradient will

E
ˆf (x)
U
be greater than that near a mode. This will allow observations far from the mode or near a local minimum to move
towards the mode faster than using ̂f E (x) alone.
The mean shift procedure obtained by successive
358
Two Robust Techniques for Segmentation of Biomedical Images

computation of the mean shift vector

translation of the window
M h, U ( x)
and
Sh (x) by M h , U ( x )
is guaranteed to converge [7, 8, 10].
2.1.1.1 Generalization
Employing the profile notation, the density estimate in the expression (3) can be written as,
 x x
fˆK (x) 
 k h i
nh d i 1 
1
n
2




(9)
The first step in the analysis of a feature space with the underlying density f(x) is to find the modes of this density.
The mode are located among the zeros of the gradient f(x) = 0 and the mean shift procedure in an elegant way to
locate the zeros without estimating the density.
By denoting with g  k  ; that is, the profile defined by the derivative of profile k with the sign changed (we
assume that the derivative of k exits x 
(see expression (4)) becomes,
n
 x  xi 2 
 x  xi
2

 h

(x
x)g

i
 nh d  2 i 1
h
nh d  2 i 1



n
 x  xi 2  

 x g
 n  x  xi 2   i 1 i  h  

g
x
i
h


1 
n  x x 2 



  g
i



 h 
i 1




ˆ f ( x )  fˆ (x) 

K
K

2
nh d  2
 x  xi 2 
g
 h 



i 1
0,  , excepting a finite set of points), then the density gradient estimate
2
n
 (x  xi )k 
2



(10)
n
where
is assumed to be nonzero.
One can observe that the derivate of the Epanechnikov profile is the uniform profile, while the derivate of the normal
profile remains as exponential.
The last bracket in expression (10) contains the mean shift vector computed with a kernel G(x) defined by
  , where c is a normalization constant; that is,
G ( x)  cg x
2
359
Roberto Rodríguez, et al.
 x  xi 2 
x
g


i 
 h 
i 1
M h,G (x)  n
x
x  xi 2 

g h 


i 1 
n
 x G
n
i 1
n
i
 G
i 1
x  xi
h
x  xi
h


x
(11)


(12)
Then, the density estimate at x becomes,
1
fˆG (x)  d
nh
 G
n
i 1
x  xi
h

c
 d
nh
n
 g 
i 1
x  xi
h
2
By using the expressions (11) y (12), the expression (10) becomes,
ˆ f ( x)  fˆ ( x)  2 M ( x)

K
G
h ,G
h 2c
(13)
ˆ f ( x)
h 2c 
K
M h , G ( x) 
2 fˆG ( x)
(14)
from where the expression,
it is a generalization of the mean shift vector (see expression (8)). This allows using other kernels, for example, Gauss
kernel, which gives wonderful results.
Basically, the algorithm that we use in this work has two steps, a filtering step and other, which is exactly the
segmentation [11]. The filtered algorithm has the following steps [8]:
Let Xi and Zi, i=1,..,n, be the input and filtered images in the joint spatial-range domain. For each pixel p  Xi,
p  ( x, y, z )  3 , where (x, y)  2 and z  [0, 2 β  1] , being the quantity of bits/pixel in the image.
1.
2.
3.
Initialization
Compute the mean shift vector until convergence. We look for the mode, in which the mentioned pixel (p)
converges. This can be carried out by using a uniform kernel o Gaussian. We used as stop criterion the
absolute error.
Assign in Zi the z component of the calculated value (intensity of the level grey).
The steps of the proposed segmentation algorithm are the following:
1.
Run the mean shift filtering procedure for the image and store all the information about the d-dimensional
convergence point in zi.
360
Two Robust Techniques for Segmentation of Biomedical Images
2.
3.
4.
5.
6.
7.
8.
Define the regions that are in the spatial domain (hs) and that its intensities are smaller or equal than hr/2 (range
domain).
For each of the defined regions in the step 2, one looks for all the pixels belonging to her and assigns in Z the
mean of its intensity values.
Build the region graph through an adjacent list of the following way: for each region, one looks for all adjacent
regions that are on the right hand side and below her.
While there exists nodes in the graph, which have been not visited, to run a variant of the “Depth-First Search”
(DFS), which concatenates adjacent regions [11], where we gave as parameter a node, which has been not
visited.
Go to the step 3 while is not obtained the stability (i. e, no more pixel value modifications).
Optional: Eliminate spatial regions containing less than M pixels, since those regions are considered irrelevant.
Optional (Binary): To all the pixels belonging to background, one assigns the white color and the black color to
objects.
2.2 Spectral Methods
As have been pointed, spectral methods are based on the calculation of the eigenvectors and eigenvalues of a matrix
derived from the affinities of the pixels in the image. Normalized Cut is a spectral method that measures both, total
dissimilarity between the different groups as well as the total similarity within groups. The main difficulty of this
method is the high dimension of the eigenproblem to be solved to natural images, which does not permit the use of
standard techniques. A possible approach in this context is considering the image as a graph only locally connected
imposing null values for the affinity between non-closed pixels, it guarantees an sparse structure for the affinity matrix
and a considerable reduction of the time and memory computational cost of the problem.
However, it is not clear how the neglected affinities affect the image segmentation and techniques that involve dense
affinity matrix promise better results. A spectral technique where all affinities are retained was proposed in [13]. The
method is an extension of the Nystrom approximation for the integral eigenvalue problem. The main idea is solving the
eigenvalue problem for a small random subset of pixels (sample) and then extrapolating this solution to the full set of
pixels.
2.2.1 Normalized cut and Nystrom method
Let G be a set of pixels and let W be a weighted affinity matrix, then the method of Normalized Cut divides to G in two
subsets G1 and G2 such that the normalized cut between them is minimum. The normalized cut is defined as,
Ncut(G1 ,G2 ) 
2cut (G1 ,G2 )
vol(G1 ) vol(G2 )
(15)
where cut(G1,G2) is the sum of the weights between G1 and G2, vol (G1) and vol (G2) is the sum of the degrees within
the sets G1 and G2, respectively, and the symbol “ || ” denotes the harmonic mean.
The solution of this minimization problem can be approached by solving the standard eigenvalue problem following:
( D 1 / 2WD 1 / 2 )V  V
(16)
where D is a diagonal matrix, Dii = j Wij and D-1/2 denotes the positive symmetric square root of the inverse of D.
Basically, the ith component of the second largest eigenvector determines the embedding of the ith pixel. This process
partitions the image in two disjointed sets.
Since the dimension N of W is the square of the G dimension, the solution of (16) involves a serious numerical
difficulty when the problem is solved by a conventional equipment. The Nystrom extension is an efficient technique that
uses the information of a subset of pixels in order to obtain an approximate solution of the eigenvalue problem. The
extension of Nystrom is an efficient technique which uses the information of a subset of pixels in order to obtain an
approximate solution of the eigenproblem.
361
Roberto Rodríguez, et al.
For the sake of simplicity, the sample is defined as the first n pixels of the image matrix. The strategy consists of
dividing the W matrix of the following way:
A
W  T
B
B
C 
(17)

B A B 
(18)
and approximating the matrix W by a low-rank matrix,
A
Wˆ   T
B
B
T
1
where the dimensions of the matrices A and B are nxn and (nx(N-n)), respectively, n<<N.
If A is a defined positive matrix, the second largest eigenvector of
column of the V matrix, which is defined of the following way,
Ŵ can be estimated by calculating the second
 A
V   T  A 1 / 2U S S1 / 2
B 
where US and
(19)
ΛS contain the spectral decomposition of the matrix,
S = A + A-1/2 B BTA-1/2
(20)
Note that the eigenproblem dimension is n, it implicates that if n is taken sufficiently small, the problem now can be
solved to a suitable computation time.
It is clear to see that an interesting aspect of the algorithm is to know when the second largest eigenvector of Ŵ is a
good approximation of the corresponding eigenvector of W. Although both vectors are theoretically different, the
T
1
structure of the affinity matrix for natural images permits that generally B A B is a good approximation for the
matrix C, and therefore the obtained segmentation by the method gives suitable results. The numerical experimentation
carried out in [13] using the Corel data base, shows that roughly 100 randomly chosen samples are sufficient to capture
the salient groups in typical natural images.
In this case, the elements of the used affinity matrix have the form,
Wij  e
Ii I j

σI
2
*e
Xi  X

σX
where Xi is the spatial localization and Ii is the gray level of the ith node.
362
2
j
(21)
Two Robust Techniques for Segmentation of Biomedical Images
3 Features of the studied images
As biomedical images, we use biopsies, which represent histological cuts in malignant tumors. These were included in
paraffin by using the inmunohistoquimic technique with the complex method of avidina biotina. Finally, monoclonal
CD34 was contrasted with green methyl to accentuate new formation of blood vessels. These biopsies were obtained
from human bodies of soft parts. This analysis was carried out for more of 80 patients. In Fig. 2 can be seen typical
images, which were captured via MADIP system with a resolution of 512x512x8 bit/pixels [14].
Fig. 2. These images represent the blood vessels in malignant tumors. Some blood vessels are marked with arrows.
In Fig. 3 can be observed a horizontal profile through the centre of a vessel; i. e., a plot of the pixel intensities along
a single row.
(a)
(b)
Fig. 3. An intensity profile through the centre of a vessel. Profile is indicated by a line in 3 (a).
There are several notable characteristics of these images, which are common to typical images that we encounter in
the tissues of biopsies:
1. The intensity is slightly darker within the blood vessel than in the local surrounding background. It is emphasized
that this observation holds only within the local surroundings.
2. High local variation of intensity is observed both within the blood vessel and the background. However, the local
variation of intensity is higher within the blood vessel than in background regions (see Fig. 2(a)).
363
Roberto Rodríguez, et al.
3.
The variability of blood vessels both, in size and shape can be observed (see Fig. 2).
Due to acquisition protocol, the images are corrupted with a lot of noise.
4 Experimental Results. Analysis and Discussion
Now, we shall expose the obtained results in the blood vessels segmentation through the mean shift and a spectral
technique, which was based on Nystrom method. In both methods the calculations were carried out by using MATLAB
(ver. 6.5). In the case of the spectral method, the computational methods were elaborated by the authors and we included
the modifications suggested in [15]. The size of the used sample was n = 30. In Fig. 4 a first example of this
comparison is represented.
(a)
(b)
(c)
Fig. 4. (a) Original image, (b) Binary image by using the mean shift vector, (c) Obtained segmentation by the spectral method.
It is possible to observe (see Fig. 4) as the two methods were robust for the segmentation of histological images (in
this case, blood vessels; but it was tested with other biomedical images). According to opinion of the physicians both
364
Two Robust Techniques for Segmentation of Biomedical Images
techniques were able of discriminating all blood vessels (BV), what it has much importance from point of view of the
diagnostic. However, it can be seen that the obtained results with the mean shift were less noisy (see Fig. 4(b)). This is
due by the effect of the used algorithm in the mean shift.
In Figs. 5, 6 and 7 appear other obtained results in the BV segmentation through the mean shift and with the spectral
method.
(a)
(b)
(c)
Fig. 5. (a) Original image, (b) Segmentation via the mean shift, (c) Obtained segmentation via the spectral method.
365
Roberto Rodríguez, et al.
(a)
(b)
(c)
Fig. 6. (a) Original image, (b) Segmentation through the mean shift, (c) Obtained segmentation with the spectral method.
366
Two Robust Techniques for Segmentation of Biomedical Images
(a)
(b)
(c)
Fig. 7. (a) Original image, (b) Segmentation via the mean shift, (c) Segmentation via the spectral method. The circles mark the
difference between the two methods.
In the presented examples, one can note as the two methods were able of isolating of effective way all the blood
vessels. However, it can be observed that the segmented images with the mean shift algorithm were less noisy than the
obtained images via the spectral method. In addition, the edges are noticed more robust and more defined in the images
segmented with the mean shift (see circles in Fig. 7). In order to obtain more clear images with the spectral method is
necessary to carry out an additional step with the goal of eliminating the spike noise which arises in the segmentation
process. This procedure could be carried out through a morphological technique, which can be an opening or a majority
filter. It is necessary to point out that this behavior was of this way for all the images which were processed.
From the point of view of algorithm, the mean shift is less complex than the spectral method and the segmentation
process was carried out faster. In fact, in a Pentium IV to 2.6 GHz, the segmentation by the spectral method took 20
minutes, while that through the mean shift delayed running only 2 minutes.
367
Roberto Rodríguez, et al.
5 Conclusions
In this work, we carried out a comparison between two robust technique of segmentation; namely, the mean shift and the
spectral methods. We demonstrated by extensive experimentation, by using real image data, as the obtained images
through the mean shift were less noisy than the spectral method. In addition, the obtained edges were more defined
through the mean shift. It was tested that the algorithm of the mean shift was 10 times more rapid than the spectral
method. These techniques can be extended to another biomedical images and its application is valid in other tasks where
it are necessary to use robust methods of segmentation.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
W. Kenong, D. Gauthier and M. D. Levine: “Live Cell Image Segmentation”, IEEE Transactions on Biomedical
Engineering, vol.42, no. 1, Jun. 1995.
J. Sijbers, P. Scheunders, M. Verhoye, A. Van der Linden, D. Van Dyck, E. Raman: “Watershed-based
segmentation of 3D MR data for volume quatization”, Magnetic Resonance Imaging, vol. 15, no. 6, pp 679-688,
1997.
C. Chin-Hsing, J. Lee, J. Wang and C. W. Mao: “Color image segmentation for bladder cancer diagnosis”, Mathl.
Comput. Modeling, vol. 27, no. 2, pp. 103-120, 1998.
Rodríguez, R., Alarcón, T., Wong, R. and Sanchez, L.: “Color segmentation applied to study of the angiogenesis.
Part I”, Journal of Intelligent and Robotic System, Vol. 34, No.1, May 2002.
P. Schmid: “Segmentation of digitized dermatoscopic images by two-dimensional color clustering”, IEEE Trans.
Med. Imag., vol. 18, no. 2, Feb., 1999.
J.E. Koss, F. D. Newman, T. K. Johnson and D. L. Kirch: “Abdominal organ segmentation using texture transforms
and a hopfield neural network”, IEEE Trans. Med. Imag., vol. 18, no. 7, July, 1999.
Comaniciu, D. I.: “Nonparametric Robust Method for Computer Vision”, Ph.D. thesis, New Brunswick, Rutgers,
The State University of New Jersey, January, 2000.
Comaniciu, D. and Meer, P.: “Mean Shift: A Robust Approach toward Feature Space Analysis”, IEEE Transaction
on Pattern Analysis and Machine Intelligence, Vol. 24, No. 5, May 2002.
Fukunaga, K. and Hostetler, L. D.: “The Estimation of the Gradient of a Density Function”, IEEE Trans.,
Information Theory, Vol. 21, pp. 32-40, 1975.
Cheng, Y.: “Mean Shift, Mode Seeking, and Clustering”, IEEE Trans., Pattern Analysis and Machine Intelligence,
Vol. 17, No. 8, pp. 790-799, Aug. 1995.
Castillo, Patricio J.: “Robust Technique for Blood Vessels Segmentation in Malignant Tumors.”, (In Spanish),
Diploma Thesis, Mathematic Faculty, Havana University, 2004.
J. Shi and J. Malik, “Normalized Cuts and Image Segmentation”, IEEE Trans. Pattern Analysis and Machine
Intelligence, vol. 22, No. 8, pp. 888-905, 2000.
C. Fowlkes, S. Belongie, F. Chung, and J. Malik, “Spectral Grouping Using the Nystrom method,” IEEE Trans.
Pattern Analysis and Machine Intelligence, vol. 26, No. 2, pp. 214-225, 2004.
Rodríguez, R., Alarcón, T and Sánchez, L.: “MADIP: Morphometrical Analysis by Digital Image Processing”,
Proceedings of the IX Spanish Symposium on Pattern Recognition and Image Analysis, Vol. I, pp. 291-298, ISBN
84-8021-349-3, 2001, Spain.
E. Izquierdo and V. Guerra, “Numerical Stability of Nystrom Extension for Image Segmentation”, Proceeding of
2004 IEEE International Workshop on Machine Learning for Signal Processing, Sept, 2004, Sao Luis, Brasil.
368
Two Robust Techniques for Segmentation of Biomedical Images
Roberto Rodríguez M. received his diploma in Physic from the Physics Faculty, Havana University in 1978 and the PhD.
Degree from Technical University of Havana, in 1995. Since 1998 he is the head of the Digital Signal Processing Group of
the Institute of Cybernetics, Mathematics & Physics (ICIMAF). His research interests include Segmentation, Restoration,
Mathematical Morphology, Visual pattern recognition, Analysis and Interpretation of images, Theoretical studies of
Gaussian Scale-Space, New Lexis on Colour via Santana Method and Mean shift. He has published more than 40 articles in
international journals and in many international conferences. He has received more of ten prizes as author and tutor.
Valia Guerra received the M.S and Ph.D degrees in Applied Mathematics from the University of Havana in 1995 and
1998, respectively. Currently, she is with the Numerical Methods Group of the Institute of Cybernetics, Mathematics and
Physics,Havana, Cuba. Her research interests include Numerical Linear Algebra, Ill-posed problems and
Regularization Methods.
Ebroul Izquierdo received the Ph.D. degree in mathematics from Humboldt University, Berlin, Germany, in 1993. He is
the head of the Multimedia and Vision Research Group, Queen Mary. His research interests include stereo vision,
efficient indexing and retrieval of video and information hidding in video signals. He is an Associated Editor of the
IEEE Transactions on Circuits and Systems for Video Technology.
369